In a digital communication system, data is transmitted to a receiver during a finite interval of time from a finite set of possible waveforms. The objective of the receiver is not to reproduce the transmitted waveform with precision, but rather to determine, from the finite set of possible waveforms, which waveform was transmitted. In the presence of noise, however, there is a finite probability of error, Pe, that the waveform detected by the receiver is not the waveform transmitted by the transmitter.
The transmitted waveform may include many forms of information, such as voice, video, data, etc., and may further be organized into groups of data bits called data symbols. The data symbols may then be used to modulate a carrier signal using a variety of modulation formats to generate the transmitted waveform. The waveform may be transmitted over a path, i.e., a channel, that may consist of a wired medium, such as a transmission line, or a wireless medium, such as a waveguide, or free space. At the receiver, both the transmitted waveform and any unwanted signals, i.e., noise, are present. The noise, being superimposed upon the transmitted waveform, tends to obscure or mask the transmitted waveform, thus limiting the receiver's ability to accurately detect the transmitted symbols.
While good engineering design may eliminate some of the undesirable effects of noise upon the received waveform, such as through filtering, channel coding, and modulation format selection, thermal noise cannot be eliminated. Thermal noise is the result of the thermal motion of electrons in all dissipative components and since thermal motion is necessary for electrical conduction, thermal noise is an unavoidable consequence.
Mathematically, thermal noise may be described as a zero-mean, Gaussian random process, whose value, x, at any time, t, is statistically characterized by the Gaussian probability density function, p(x), of equation (1):
                                          p            ⁡                          (              x              )                                =                                    1                                                σ                  x                                ⁢                                                      2                    ⁢                    π                                                                        ⁢                          exp              ⁡                              [                                                                            -                      1                                        2                                    ⁢                                                            (                                              x                                                  σ                          x                                                                    )                                        2                                                  ]                                                    ,                            (        1        )            where σx2 is the variance of x. Graphically, the Gaussian probability density function of equation (1) yields the familiar “bell-shaped” curve.
Given, for example, that a transmitted symbol of the digital transmission system may take on one of two binary states, e.g., a1 and a2, such as is the case for a binary phase shift keying (BPSK) signal, the superposition of the Gaussian probability density function of equation (1) with the transmitted symbols, a1 and a2, yields the conditional probability density functions (CPDF) of FIG. 1. CPDF 102 may be expressed as in equation (2) as:
                              p          ⁡                      (                          z              /                              a                1                                      )                          =                              1                          σ              ⁢                                                2                  ⁢                  π                                                              ⁢                                    exp              ⁡                              [                                                                            -                      1                                        2                                    ⁢                                                            (                                                                        n                          -                                                      a                            1                                                                          σ                                            )                                        2                                                  ]                                      .                                              (        2        )            Similarly, CPDF 106 may be expressed as in equation (3) as:
                              p          ⁡                      (                          z              /                              a                2                                      )                          =                              1                          σ              ⁢                                                2                  ⁢                  π                                                              ⁢                                    exp              ⁡                              [                                                                            -                      1                                        2                                    ⁢                                                            (                                                                        n                          -                                                      a                            2                                                                          σ                                            )                                        2                                                  ]                                      .                                              (        3        )            
Thus, the area under CPDF 102 to the left of the threshold level 114 represents the probability that the receiver's detected output is equal to 104, given that symbol a1 was transmitted. Likewise, the area under CPDF 106 to the right of the threshold level 114 represents the probability that the receiver's detected output is equal to 108, given that symbol a2 was transmitted. Overlapping areas 110 and 112 represents the likelihood that the receiver's detected output is incorrect for a given transmitted symbol. That is to say, for example, that area 110 to the left of threshold level 114 under CPDF 106 represents the probability that symbol a1 is detected, given that symbol a2 is transmitted. Similarly, area 112 to the right of threshold level 114 under CPDF 102 represents the probability that symbol a2 is detected, given that symbol a1 is transmitted.
Overlapping areas 110 and 112 represent relatively low probabilities of error in high SNR environments. In a low SNR environment, on the other hand, the area of overlapping portions 110 and 112 may increase dramatically, thereby increasing the probability that the wrong symbol is detected for a given transmitted symbol.
Low SNR environments also induce other detrimental effects upon the receiver of a digital communication system. In particular, all digital communication systems require at least some degree of synchronization, so that incoming signals may be coherently detected at the receiving end. For example, carrier phase/frequency synchronization at the receiver is often required for coherent reception. In many instances, symbol phase/frequency synchronization may also be required for coherent reception. Low SNR environments, however, impede the receiver's ability to “track” the phase/frequency of the received carrier signal and the received symbols, thus increasing the probability of error above what is theoretically achievable.
Thus, an adaptive synchronization control mechanism may help to improve the phase/frequency tracking performance of the receiver. In particular, the loop parameters of a phase locked loop (PLL), or a frequency locked loop (FLL), implemented within the receiver, may be adjusted in response to an SNR estimate, so that, for example, the loop bandwidth of the PLL, or FLL, may be optimized for a given SNR estimate.
Unfortunately, the accuracy of prior art SNR estimates degrades with decreasing SNR. Thus, the ability to adaptively improve the phase/frequency tracking performance of a receiver in low SNR conditions is decreased due to the inaccuracy of the prior art SNR estimate. Efforts continue, therefore, to improve the accuracy of SNR estimation at low SNR, so that among other receiver functions, synchronization performance in low SNR conditions may be improved. Other receiver functions, such as automatic gain control (AGC) loops, may also benefit from improved SNR estimates at low SNR, in order to allow the gain value of the AGC loop to adapt to the estimated SNR.